Simulated Annealing and Genetic Algorithms for the ... - Springer Link

Computational Optimization and Applications, 7, 111–126 (1997)

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Simulated Annealing and Genetic Algorithms for the Facility Layout Problem: A Survey THELMA D. MAVRIDOU, AND PANOS M. PARDALOS

[email protected]

Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611-6595 USA

Abstract. The facility layout problem (FLP) has many practical applications and is known to be N P-hard. During recent decades exact and heuristic approaches have been proposed in the literature to solve FLPs. In this paper we review the most recent developments regarding simulated annealing and genetic algorithms for solving facility layout problems approximately. Keywords: Heuristics, Simulated Annealing, Genetic Algorithms, Facility Layout Problem, Parallel Algorithms, Combinatorial Optimization.

1.

Introduction

The facility layout problem deals with the physical arrangement of a given number of departments or machines within a given configuration. In the context of manufacturing the objective is to minimize the total material handling cost of moving the required material between the departments. The importance of material handling is stated by Tompkins and White [67] who claimed that 20-50 % of the total operating expenses within manufacturing are attributed to it. The facility layout problem is one of the best-studied problems in the field of combinatorial optimization. A number of formulations have been developed for the problem. More particularly the FLP has been modeled as [36] : 1. quadratic assignment problem (QAP), 2. quadratic set covering problem, 3. linear integer programming problem, 4. mixed integer programming problem, 5. graph theoretic problem. The quadratic assignment formulation has been traditionally used to model the facility layout problem. QAP was first introduced by Koopmans and Beckmann [33] in 1957 as a mathematical model for locating a set of indivisible economic activities. Consider the problem of allocating a set of facilities to a set of locations, with the objective to minimize the cost associated not only with the distance between locations but with the flow also. More specifically, given two n × n matrices F = (fij ) and D = (dkl ) where fij is the flow

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between the facility i and the facility j, and dkl is the distance between the location k and the location l, and a set of integers N = {1, 2, . . . , n}, the QAP can be written as follows: min

p∈ΠN

n X n X i

fij dp(i)p(j)

(1)

j

where ΠN is the set of all permutations of N , and n is the number of facilities and locations [56]. Exact algorithms for solving the FLP include branch and bound ([39], [30]) and cutting plane algorithms ([7], [9]). These approaches require rather high computational time as the problem size increases, resulting in practice in the solution of only moderately sized problem instances. Therefore, a number of heuristic algorithms, such as construction, improvement and hybrid algorithms, have been developed for sub-optimally solving large-size instances in a reasonable amount of CPU time and computer memory. Recent survey papers on the facility layout problem and its solution approaches can be found in [22], [36], and in [43]. In this paper we focus on the work that has been done to date for solving the facility layout problem using simulated annealing (SA) and genetic algorithms (GA). Both heuristic approaches are stochastic search techniques modeled on processes found in nature (thermodynamic process and natural evolution). These heuristic methods have been used to solve a wide variety of combinatorial optimization problems ([12], [19], [37], [47]). 2.

Simulated Annealing for the Facility Layout Problem

Simulated annealing was first proposed by Kirkpatrick et al. [31] as a method for solving combinatorial optimization problems. The name of the algorithm derives from an analogy between the simulation of the annealing of solids first proposed by Metropolis et al. [44], and the strategy of solving combinatorial optimization problems. Annealing refers to a process of cooling material slowly until it reaches a stable state. Starting from an initial state, the system is perturbed at random to a new state in the neighborhood of the original one, for which a change of ∆E in the objective function value (OFV) takes place. In a minimization process if the change ∆E is negative then the transformation to the new state is accepted. −∆E If ∆E ≥ 0 the transformation is accepted with a certain probability of p(∆E) = e kb T , where T is a control parameter corresponding to the temperature in the analogy and kb is Boltzmann’ s constant. The change ∆E in the OFV corresponds to the change in the energy level (in the analogy) that occurs as the temperature T decreases. SA gives us a mechanism for accepting small increases in the objective function value, controlling though the probability of acceptance p(∆E) through the temperatures. Kirkpatrick et al. [31] argue that allowing “hill climbing” moves, one can avoid configurations that lead to locally optimal solutions and eventually higher quality solutions can be obtained. So the main advantage of the simulated annealing method is its ability to escape from local optima. The main features of the SA method are [14] : •

the temperature T , which is the parameter that controls the probability p(∆E) of accepting a cost-increasing interchange. During the course of the algorithm T is decreased

SA AND GA FOR THE FACILITY LAYOUT PROBLEM: A SURVEY

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in order to steadily reduce the probability of acceptance of interchanges that increase the value of the objective function, •

the equilibrium, i.e. the condition in which a further improvement in the solution using additional interchanges is highly unlikely to occur,

•

the annealing schedule that determines when and by how much the temperature is to be reduced.

A pseudo-code of the simulated annealing procedure is given in Figure 1 [54].

Simulated Annealing Procedure Input: A problem instance Output: A (sub-optimal) solution 1. Generate an initial solution at random and initialize the temperature T . 2. while (T > 0) do (a) while (thermal equilibrium not reached) do (i) Generate a neighbor state at random and evaluate the change in energy level ∆E. (ii) If ∆E < 0 update current state with new state. (iii) If ∆E ≥ 0 update current state with new state −∆E

with probability e kb T , where kb is a constant. (b) Decrease temperature T according to annealing schedule. 3. Output the solution having the lowest energy.

Figure 1. Simulated Annealing Procedure

Several implementations of the simulated annealing algorithm have been proposed for the facility layout problem. We will present the main concepts of the most recent approaches and comment on the computational results. Heragu and Alfa in [21], present an extensive experimental analysis of two simulated annealing based algorithms, implementing them on two patterns of layout, the single-row

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and multi-row facility layouts. The first algorithm uses the standard techniques of the SA heuristic. In the main step the algorithm examines the random exchange of the positions of two facilities. The new solution is accepted if the exchange results in a lower OFV. Otherwise, the difference ∆E between the OFV of the best solution obtained so far and −∆E the current solution is computed. This solution is accepted with probability e T . This step is repeated 100n times or until the number of new solutions accepted is equal to 10n, where n is the number of facilities in the layout problem. Next, the algorithm decreases the value of temperature T by multiplying it by the cooling ratio r and repeats the main step. The stopping criterion is a fixed maximum number of temperature change steps. The initial temperature T is set as a number sufficiently larger than the largest ∆E encountered for problems tested with other heuristics. Guidelines for setting the parameters can be found in [57]. The second algorithm presented in the same paper is a hybrid SA algorithm (HSA), which uses a “core” algorithm to generate a “good” initial solution, and then improves it using the SA algorithm described before. The core algorithm is a modified penalty algorithm (MP) presented in [22]. Eight test problems of size up to 30 (available in the literature) are used for the single-row case. Each test problem is solved 10 times using the same initial solution. For six of the problems the HSA algorithm produces optimal or best-known solutions. For the remaining two problems, the solutions are better than those previously reported in the literature. A comparison between the HSA and the SA algorithms is presented, as well as with three other heuristic algorithms (a 2-way exchange, a 3-way exchange and a WilhemWard version of simulated annealing [70]) using 15 equal-area multi-row FLPs. The HSA in terms of solution quality, performed better than all the other algorithms though requiring more computational time than the SA algorithm. Also as the number of annealing runs increases, SA seems to produce similar quality solutions with HSA with less computational effort. Another implementation of the SA algorithm applied to the cellular layout problem can be found in [27]. This problem involves the determination of the relative positions of n equidimensional manufacturing entities which may represent either the set of machines belonging to a cell (intra-cell problem) or the manufacturing cells within a shop (inter-cell problem). The objective of both layout problems is to minimize the total material flow (cost) between the manufacturing entities. The method presented in the paper is called CLASS, which stands for Computerized LAyout Solutions using Simulated annealing. The proposed algorithm is a regular simulated annealing algorithm with the following most important elements: •

Solution space: The solution space consists of a n × n grid, i.e. n2 positions are available to be occupied by the n entities. The distance between all pairs of positions is determined using geometric or Manhattan distances.

•

Interchanges: The interchange given a solution can be either a move of an entity from its current position to an unoccupied position or an “exchange” of the positions of two entities. The two positions from the solution space that are exchanged are selected randomly.


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•

Annealing schedule: The annealing schedule considers the initial temperature to be sufficiently large so that all interchanges are eventually accepted. The temperature is reduced by multiplying it with a constant that takes values between 0 and 1.

•

Parameters: The number of interchanges to be attempted at each temperature, the number of accepted interchanges at each step and the total number of temperature change steps are 100n, 10n and 100 respectively, following the guidelines of [57], exactly as in [21].

•

Interchange Acceptance Criterion: The interchange is accepted if a randomly generated −∆E number between 0 and 1 is less than the value of e T , where ∆E and T are respectively the difference in the OFV and the temperature at the current step.

CLASS was compared to twelve other layout methods (discussed in [36]) in terms of both the quality of the solution and the speed of convergence. Eight problems available in the literature were used for the comparison of the algorithms, with sizes between n = 5 and n = 30. In each case CLASS either equals the performance of, or outperforms each of the other methods. The sensitivity of CLASS to the initial conditions was tested by running each of the test problems of sizes 5,6,7, and 8, five times, each time with a different initial solution. The optimal solution was obtained in each case, indicating the insensitivity of the solution quality to the initial conditions. For the inter-cell problem Tam in [64] describes a SA solution approach which takes into consideration the traffic between cells, the geometric constraints of the individual cells and any occupied regions on the floor plan. The objective is to find a layout that minimizes the weighted flow of parts between the manufacturing cells while satisfying the area and shape constraints of the individual cells. There are several critical points concerning the problem formulation: •

Layout representation: The layout takes the form of a slicing structure, which is represented by a slicing tree. This is a binary tree representing the recursive partitioning process of a rectangular area, through cuts. A cut specifies the relative position of the departments (left, right, below or above each other) through four distinguished branching operators.

•

Solution space: The solution space is defined as the set S which consists of all slicing trees that can be generated by rearranging cuts of a given structure. It is shown that |S| = 4n−1 , where n is the number of cells and the size of the neighborhood N is |N | = 4n − 5.

•

Area constraints: The location where a rectangular partition is cut, i.e. the cut point, must be chosen so that the split partitions receive their required areas. The cut point is determined in a top-down fashion starting from the “root” of the tree.

•

Shape constraints: The cell’ s shape is described using the aspect ratio and the deadspace ratio. The first ratio is the height over the width of the partition allocated to a cell. The second ratio is used to measure the amount of unusable space within the partition allocated to a manufacturing cell. Both ratios have lower and upper bounds.

116 •


Slicing tree construction: Using numerical clustering techniques a slicing tree is constructed in such a way that cells with large inter-cell traffic volume are placed in close proximity with each other.

The attractive element of the algorithm is that it exploits the hierarchical representation of the layout, so that the probability of selecting a neighborhood state is not uniformly distributed (as in a regular SA algorithm), but is dependent on T . More particularly, when T is high at the first steps of the annealing procedure, a cut near the root of the slicing tree will be selected, causing large swings in the cost function value since a large number of cells will have to be relocated. As T decreases during the course of the algorithm, cuts that are located at a lower level in the tree are selected, to generate a neighborhood state. So a guided search in the set of neighboring solutions is adopted. The algorithm was compared to two other local search methods, denoted as HC (a straightforward hillclimbing method) and BC (a modified version of HC). Two test problems of size n = 20 and n = 30 were constructed for the comparison. Each method was run 10 times with different initial solutions. The computation time was kept the same among the three methods. In terms of solution quality the proposed SA algorithm outperformed the other two methods, both in average and minimum cost. In [34] Kouvelis and Chiang address the single row layout problem (SRLP) in flexible manufacturing systems (FMSs). The problem deals with the optimal arrangement of n machines along a straight track with a material handling device moving jobs from one machine to another. The difficulty of the problem is due to the variety of parts to be processed in different ranges of operation sequences. When the sequence of operations of a job is not the same as the sequence of the locations of the machines, the job sometimes has to travel in reverse (backtrack) in order to receive the required operations. The objective of the SRLP is to find the ordering of the machines that minimizes the total backtracking distance of the material handling device. If we consider n machines and n candidate locations for the machines to be placed, the solution to the SRLP is one of the possible permutations of the set S = {1, 2, . . . , n} defined as the set of the workstation assignment vectors, each one representing a configuration of the machines in a single row. The neighborhood of a configuration is the set N of configurations resulting by the interchange of the locations of two machines. The initial configuration is obtained by randomly assigning machines to locations. For the setting of the parameters of the SA algorithm, i.e. the initial acceptance probability (through which the initial temperature will be calculated), the number of interchanges attempted before the reduction of the temperature, the value of the cooling ratio, and the number of steps to reach the equilibrium, a sensitivity analysis was performed with respect to each individual parameter. For each parameter a range of values is tested while all other parameters are held fixed. The best values of the parameters are kept as the final ones to be used in the algorithm. The experimental analysis showed that fine-tuning of the SA parameters with respect to each specific application and the selection of the initial solution is very important for the performance of the algorithm in terms of quality solution. The same authors and J. Fitzsimmons in [35] describe two distinct implementations of the simulated annealing algorithm for machine layout problems in the presence of zoning constraints. These constraints are restrictions on the arrangement of machines. Positive zoning constraints require that certain machines have to be placed near each other, while


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negative zoning constraints do not allow certain machines to be in close proximity. The problem is formulated as a restricted quadratic assignment problem. Assuming that the number of candidate locations is equal to the number of machines, the objective is to assign the machines to the locations in a way that the cost function is minimized with respect to the zoning constraints. The first of the SA algorithms called the Compulsion Method, takes into consideration the zoning constraints mostly during the search for a new layout in the neighborhood of the original one. The second algorithm, the Penalty Method, takes into account the presence of the zoning constraints in the objective function through the use of appropriate penalty terms. For each layout that violates any of the zoning constraints, corresponding penalty terms are charged in the OFV. The fine-tuning of the parameters for both SA procedures and the interpretation of the configuration, the neighborhood of a configuration and the initial configuration are the same as described for [34]. The two versions are compared on an extensive set of computational experiments using test problems of size ranges from 5 to 30 machines. The results showed that the Compulsion Method outperforms the Penalty Method in terms of CPU time and solution quality. The basic advantage of the Penalty Method is that it can be easily changed to handle the addition of extra zoning constraints. Meller and Bozer in [42] describe a Simulated Annealing Based Layout Evaluation algorithm (SABLE), which introduces a new generator routine for candidate layout solutions, combined with the use of spacefilling curves. The algorithm is implemented on a set of single and multiple floor facility layout problems. For the single-floor case test problems of sizes 11 to 25 are used, and the performance of SABLE is compared to the performance of the algorithms presented in [2], [49], [70], and [8]. An average and a worst-case analysis shows that the proposed algorithm performs the best in terms of solution quality. Additionally, SABLE performed better than Tam’s SA algorithm [64] on a data set of 20 and 30-size department single-floor FLPs. Let us note that regarding the department shapes, Tam’s algorithm generally assumes rectangular shapes, while the proposed algorithm tends to generate departments with non-rectangular shapes. For the multi-floor case, test problems with up to 4 floors and 40 departments were used to evaluate the performance of SABLE. The results indicate the robustness of the algorithm to changes in the vertical to horizontal ratio. Other recent Simulated Annealing algorithms for layout problems can be found in [62] and [61]. For the special case of QAP several SA approaches have been proposed. Burkard and Rendl [10] were the first to apply simulated annealing for solving the QAP. They reported on rather favorable computational results indicating that the obtained solutions deviate only 1 − 2 % from the best known solutions. Wilhelm and Ward [70] also applied the SA algorithm to quadratic assignment problems, by further experimenting on the procedure. They report on the sensitivity of SA to the control parameters, and evaluate the algorithm using problems ranging in size from n = 5 to n = 100. In particular computational results were provided for the test problems in Nugent et al. [49] and for two test problems they introduced in the paper. In [11], Connolly discusses the implementation of SA on 7 problems. The computational results indicate that examining sequentially generated neighboring solutions, rather than randomly generated ones, makes the SA algorithm more

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efficient. In [59] and [60] simulated annealing is used as a tool for interactive facility layout decisions. More recently Laursen [38] investigated the performance of the SA algorithm by varying two parameters: (1) the number of simulations, and (2) the simulation length, while in both cases the algorithm uses the same computational time for a specific instance problem. Laursen concluded that the length of each simulation is optimizable and that a large range of its values generate a near-optimal solution quality. 3.

Genetic Algorithms for the Facility Layout Problem

Genetic Algorithms (GA) were first introduced by John Holland et al. [23] at the University of Michigan in 1975. Genetic algorithms are search algorithms based on the mechanics of natural selection and natural genetics. GA try to imitate the development of new and better populations among different species during evolution. Unlike most of the heuristic search algorithms, GA conduct the search through the information of a population consisting of a subset of individuals, i.e. solutions. Each solution is associated with a fitness value, which is the objective function value of the solution. Solutions to optimization problems can often be coded to strings of finite length. The genetic algorithms work on these strings. The encoding is done through the structure named chromosomes, where each chromosome is made up of units called genes. There are some determining factors that strongly affect the efficiency of genetic algorithms : 1. The representation of the solutions by strings. 2. The generation of the initial population. 3. The selection of individuals in an old population (parents) that will be allowed to affect the individuals of a new population. 4. The genetic operators that are used to recombine the genetic heritage from the parents to produce children. The most often-used operators are the crossover and the mutation. The selection of individuals that will be allowed to affect the following generation is based on the fitness of the individuals. This is done in such a way that individuals with better fitness are more likely to be chosen to become parents. The recombination of the population consists of the following four operations: •

Crossover. By combining the coded solution strings of two parents two children are created. If one considers the biological origin of the genetic algorithms it makes sense to denote the coded solution string “genome” and look at this procedure as a result of mating. To avoid chaotic behavior, not all individuals in the new population are generated by this operator. The probability of applying this operator (crossover rate) is denoted by pc .

•

Mutation. In order to give the populations new impulses some random changes in the genomes are allowed to occur. The mutation operator changes a “gene” in a solution with a probability (mutation rate) pm .


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•

Local search. It has proven very efficient to search for locally optimal solutions in the neighborhood of the children [40]. If one is able to find a better solution then it will replace the original child as a member of the new population.

•

Control of new individuals. It is not unlikely that a child will have worse fitness than its parents. In that case the child might not be accepted in the new generation.

Let us note also that a GA implementation requires the specification of certain parameters such as population size, and number of generations. Let Pt denote the population at time t. Then the genetic algorithm procedure can be described as in Figure (2) [54].

Genetic Algorithm Procedure Input: A problem instance Output: A (sub-optimal) solution 1. t = 0, initialize Pt , and evaluate the fitness of the individuals in Pt 2. while (termination condition is not satisfied) do (a) t = t + 1 (b) Select Pt , recombine Pt and evaluate Pt 3. Output the best solution in the population as the (sub-optimal) solution.

Figure 2. Genetic Algorithm Procedure

We continue with the description of various implementations of the genetic algorithm for the facility layout problem. As we have seen in the section of SA for the facility layout problem, Tam [64] uses a simulated annealing approach to solve the inter-cell problem. The same author using the same problem formulation and representation of the floorplan layout as a slicing tree, attempts a solution approach to the problem using Genetic Algorithms [63]. In applying a GA an important part of the implementation is the coding of solutions as strings of finite length. For the problem formulation under consideration, a slicing tree can be generated by a string using as its elements the nodes of the tree in a sequence which starts from the bottom level nodes and ends at the root of the tree. The nodes of the tree represent either facility identifications (operands) or “cut” symbols (operators). The proposed GA uses for the recombination of the population the crossover and mutation operators, as described for

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the general genetic algorithm. For the selection of the new population the reproduction operator is used. Under this operator the chance of being selected to remain in the new population Pt+1 is proportional to the fitness value of the individual. This operator assigns to each individual a sampling rate T (s, t) = µ(s)/¯ µ(s), where function µ measures the fitness level of individual s and µ ¯(s) is the average fitness of Pt . So the individuals with above the average fitness will have a higher survival probability than those below the average fitness. The selection of the parameters population size, crossover rate pc and mutation rate pm , is based on previous studies that can be found in the literature ([18], [58]). Four layouts with 12, 15, 20, and 30 facilities were retrieved from Nugent et al. [49]. Initial solutions were obtained by randomly generating cut operators for 30 slicing trees. For each case the GA was run for 150 generations with 10 different sets of initial solutions. The best and average solution in each generation were gathered. The performance of GA was compared with that of a hillclimbing method (HC), which searches through a neighborhood N , where N is the set of operator sequences generated from changing one operator. GA outperformed HC both in terms of minimum and average costs. For the 30-facility layout GA improved the minimum cost by 10.5% and the average cost by 13%. Koakutsu and Hirata [32] propose an interesting combined approach called genetic simulated annealing (GSA) for the solution of the floorplan design of VLSI (Very Large Scale Integrated) circuits. The problem involves the arrangement of a given set of rectangular modules (with no fixed shapes or dimensions) in the plane, with the objective to minimize: (1) the area of the enclosing rectangle which should contain all the modules, and (2) the total wire length between modules that should be connected in the circuit. The main features of the algorithm are the following: •

Stochastic Optimization: GSA uses the stochastic optimization used in simulated annealing so that a neighbor state for which there is an increase of the cost function is accepted with a certain probability.

•

Multiple Search Paths: A population of solutions corresponding to the population of GAs is used to initialize the search in multiple directions. The stochastic optimization is applied to each solution of the population.

•

Selection of search paths: The selection operator replaces solutions which have value higher than the average value of the population, with solutions that have lower cost value than the average value of the population. This way, paths which are expected to reach good solutions are selected.

•

Genetic Operators: A genetic crossover operator is used to generate new solutions.

The formulation of the problem represents the floorplan layout as a slicing tree. The representation of a solution as a string is similar to the one described previously in [63], using in this case, vertical and horizontal cuts with corresponding branching operators. GSA is tested on three floorplan problem instances. The first has 16 modules, each one a fixed square of unit area, having wires connecting to its horizontal and vertical neighbors. The second problem has 16 modules and 25 wires, and the third one has 20 modules and 31 wires. For the last two problems the total module area is 100. The proposed algorithm


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was compared to a regular SA algorithm. Both algorithms run 100 times with different initial solutions for each of the above problem instances. The average costs are used for the comparison. The results show that GA improves the average cost by 1.7% − 9.8% compared to the SA within the same computational time. More recently Banerjee and Zhou [3] developed a genetic algorithm to solve a variation of Montereuil’ s mixed integer programming formulation for the FLP [46], and in particular for the special case of single loop material flow path configuration. They introduce a “knowledge-augmented mutation operator” to determine the flow path direction, which appears to perform well for the cases where the layout has very low flow path dominance. Previous applications of GA for facilities layout design can be found in [4] from the same authors and Montreuil. Tate and Smith [65] applied GA using an adaptive penalty function to the unequal-area facility layout problem with shape constraints. The shape restrictions are expressed through a flexible bay structure proposed in the literature [68]. The rectangular area in which the facilities are to be located is divided into vertical bays of different width and each bay is divided into rectangular departments of different length. The encoding of the solutions to strings is done with two distinct chromosomes. The first one is the sequential chromosome which is represented by a permutation of the set N = {1, 2, . . . , n}, where n is the number of departments. The sequence of the permutation starts by reading departments bay to bay, from top to bottom and from left to right at the rectangular area. The second chromosome is the bay chromosome where each gene shows for each bay the number of departments contained in the previous bays including the involved one, showing this way the breaks that occur in the sequence between bays. For example, consider 4 bays having 3, 4, 6 and 2 departments respectively starting from the left bay. Then using the bay chromosome the solution encoding is (3, 7, 13). Note that the last breakpoint at 15 is obvious. The proposed GA uses variants of crossover and mutation operators. •

The variant of the crossover operator works as follows: using two individuals to be the parents, one offspring (child) is generated by the following rules. For the case of GA encoding using the sequential chromosome, each location in the child’ s sequence is the department number in the corresponding location from one of the parents, both having the same probability to be selected. This will force the common locations in the sequences of the parents to be carried over to the child. Also each department must occur only once in the child. For the bay chromosomes, the location and number of bay breaks in the child’ s sequence is taken from one of the parents, both having equal probabilities to be selected.

•

The mutation uses three different operators. Two of the operators alter the number of bays affecting only the bay chromosome and one operator reverses a subsequence of the departments affecting the sequence chromosome.

The evolution parameters, i.e. the population size, and the crossover and mutation rates are determined after several trial runs. An adaptive penalty function is used to find good feasible solutions. The penalty function is adaptive because during the course of the algorithm it uses observed population data to adjust the level of the penalty that is applied to the infeasible solutions. Test problems with size ranges from 10 to 20 departments, already published in

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the literature ([6], [69], [2]) were used to evaluate the efficiency of the proposed genetic algorithm. The proposed approach proved to be the best in terms of quality solution when compared with previous published results for the problems under consideration. Genetic algorithms are inherently parallel in nature. Several implementations of GA in parallel environments have recently appeared, introducing in this way a new group of GA, the Parallel Genetic Algorithms (PGA). The population of a parallel genetic algorithm is divided into subpopulations. Then an independent GA is locally performed on each of these subpopulations, and the best solutions in each case are transferred to all the other subpopulations. Two types of communication are established among the subpopulations [48]. Either among all nodes where the best solution of each subpopulation is broadcasted to all the other subpopulations, or among the neighboring nodes, where only the neighboring subpopulations receive the best solutions. The most important features of PGA, which result in a considerable speedup relative to sequential GAs, are the following [28]: •

Local selection : In sequential GAs the selection operation takes place by considering the whole population. In a PGA this operation is performed locally by the selection of an individual in a neighborhood.

•

Asynchronous behavior : It allows the evolution of different population structures at different speeds, resulting in an overall improvement of the algorithm in terms of computational time.

•

Reliability in computation performance : The computation performance of one processor does not affect the performance of the other processors.

Several implementations of PGA have been proposed for the solution of the quadratic assignment problem. An application of an asynchronous parallel GA called ASPARAGOS has been presented by Muhlenbein [47] for the QAP, introducing a polysexual voting recombination operator. The PGA was tested on QAPs of size 30 and 36 with known solutions. The algorithm found a new optimum for the Steinberg’s problem (QAP of size 36). The numbers of processors that were used to run this problem were 16, 32 and 64. The 64 processors implementation (on a system with distributed memory) gave by far the best results in terms of computational time. Furthermore, Huntley and Brown [26] developed a parallel hybrid of SA and GA to solve the QAP approximately. A parallel genetic algorithm is used to produce a good initial solution for each population and the SA algorithm is used for improving these solutions. More recently, Battiti and Tecchiolli in [5] developed parallelization schemes of genetic algorithms for quadratic assignment problems presenting indicative experimental results. 4.

Concluding Remarks

In this paper we summarized the work that has been done in recent years in implementing simulated annealing and genetic algorithms for solving the facility layout problem. Both heuristic approaches have been successfully used to approximately solve difficult combinatorial optimization problems. For the FLP also, the procedures seem to find sub-optimal


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solutions in a reasonable amount of computational time. Considering the latest interest and experience in efficient implementation of search algorithms on parallel multiprocessing computers ([41], [50], [51], [52], [54], [53], [55]) that significantly increase the sizes of the problems that can be solved, the use of these heuristics becomes more attractive. It is expected that efficient parallel implementations of simulated annealing and genetic algorithms will be very useful for practitioners dealing with facility layout problems. 5.

Acknowledgments

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